Unit 7: Oscillations

20 terms. Say what each one means, then open the card to check.

• Equilibrium position

  The position where the restoring force on an oscillating object is zero (for a mass on a level spring, the whole net force is zero there). The object moves fastest through it and is always pulled back toward it.

  Example. A wind chime hanging still marks its equilibrium position; a gust pushes it aside and it swings back through that spot.

• Restoring force

  A net force that always points back toward the equilibrium position, opposite to the displacement.

  Example. A trampoline mat pressed down by a jumper pushes up, back toward its flat resting shape.

• Simple harmonic motion

  Back-and-forth motion in which the restoring force, and so the acceleration, is proportional to the displacement from equilibrium and points the opposite way.

  Example. For a mass on an ideal spring, pulling it twice as far from equilibrium makes the spring pull back twice as hard.

• Hooke’s law

  An ideal spring pushes or pulls with a force proportional to how far it is stretched or compressed from its relaxed length: $F_s = -k\Delta x$.

  Example. A spring with k = 300 N/m squeezed 2.0 cm shorter than its relaxed length pushes back with 6.0 N.

• Spring constant

  The stiffness k of a spring, in N/m: the force needed for each meter of stretch or compression. A stiffer spring has a larger k.

  Example. If a 9.0 N pull stretches a spring by 0.060 m, its spring constant is 150 N/m.

• Amplitude

  The greatest distance an oscillating object gets from equilibrium, measured from equilibrium to one turning point, not from end to end.

  Example. A cart that moves between marks 10 cm to the left and 10 cm to the right of equilibrium has an amplitude of 10 cm, not 20 cm.

• Small-angle condition

  A pendulum’s restoring force is $mg\sin\theta$, which is close to $mg\theta$ (θ in radians) only for small angles, so its motion is nearly simple harmonic only for small swings.

  Example. At 10°, $mg\theta$ is only about 0.5 percent larger than $mg\sin\theta$; at 45° it is about 11 percent larger.

• Period

  The time for one complete cycle of an oscillation, out and back to the same point moving the same way, in seconds.

  Example. A porch swing that goes out and back once every 3.2 s has a period of 3.2 s.

• Frequency

  The number of complete cycles each second, in hertz (Hz); it is the reciprocal of the period, $f = 1/T$.

  Example. Wings that beat 50 times each second have a frequency of 50 Hz and a period of 0.020 s.

• Period of a mass on a spring

  $T = 2\pi\sqrt{m/k}$: a heavier mass or a softer spring gives a longer period. Neither the amplitude nor g changes it.

  Example. Swapping in a spring four times as stiff, with the same mass, halves the period.

• Period of a simple pendulum

  $T = 2\pi\sqrt{L/g}$ for small swings: a longer string or weaker gravity gives a longer period. The bob’s mass does not change it.

  Example. A pendulum 1.0 m long on Earth has a period of about 2.0 s.

• Period and amplitude

  For a mass on a spring, and for a pendulum at small angles, the period does not depend on the amplitude: a bigger swing moves faster and covers its longer path in the same time.

  Example. As a pendulum’s small swing slowly shrinks, each swing still takes the same time.

• Hanging (vertical) spring

  Hanging a mass from a vertical spring moves the equilibrium position down by mg/k, but the period is the same as for the same mass and spring lying flat.

  Example. A 2.0 kg bag on a 400 N/m spring rests 4.9 cm below the spring’s relaxed end and bounces with the same period it would have on a frictionless table.

• Position–time graph of an oscillator

  A smooth repeating cosine or sine curve. Its highest and lowest values are +A and −A, and the time between matching peaks is the period.

  Example. A cart released from rest at +5 cm is at +5 cm at t = 0, at 0 at a quarter period, and at −5 cm at half a period.

• Velocity in simple harmonic motion

  The velocity is zero at the turning points and largest in size at equilibrium; on a position–time graph it is the slope.

  Example. On a swing you hang still for an instant at the top of each arc and move fastest at the bottom.

• Acceleration in simple harmonic motion

  The acceleration along the path always points toward equilibrium and is proportional to the displacement: largest at the turning points, zero at equilibrium.

  Example. A cart at its far turning point is at rest for an instant, yet that is where the spring gives it its biggest acceleration back toward the center.

• Energy of a spring oscillator

  With no friction, the total mechanical energy $E = \frac{1}{2}kA^2$ stays fixed, and at every position $\frac{1}{2}kx^2 + \frac{1}{2}mv^2 = \frac{1}{2}kA^2$.

  Example. A 500 N/m spring with an amplitude of 0.040 m stores 0.40 J at each turning point, and the mass has 0.40 J of kinetic energy as it passes the center.

• Maximum speed

  At equilibrium all the energy is kinetic, so $v_{max} = A\sqrt{k/m}$; doubling the amplitude doubles the maximum speed.

  Example. A 2.0 kg mass on a 50 N/m spring with an amplitude of 0.30 m reaches 1.5 m/s at the center.

• Energy in a pendulum

  A pendulum trades gravitational potential energy at the ends of its swing for kinetic energy at the bottom: $mgh = \frac{1}{2}mv^2$ when nothing removes energy.

  Example. A bob released 0.20 m above its lowest point reaches about 2.0 m/s at the bottom.

• Damping

  The gradual loss of mechanical energy to friction or air drag, which makes the amplitude shrink over time.

  Example. A playground swing left alone slowly stops, because air drag and friction at the chains take energy away each cycle.